How to Analyze Data From Cluster Samples

Different sampling methods
use different formulas to estimate population
parameters and to estimate
standard errors. The formulas that we have used so far in this tutorial
work for simple random samples and for stratified samples, but they are not
right for cluster samples.

The next two sections of this lesson show the correct formulas to use with
cluster samples. With these formulas, you can readily estimate population
parameters and standard errors. And once you have the standard error, the
procedures for computing other things (e.g.,
margin of error,
confidence interval, and
region of acceptance) are largely the same for cluster samples as for
simple random samples. The sample problem at the end of this lesson shows
how to use these formulas to analyze data from cluster samples.

Measures of Central Tendency

The table below shows formulas that can be used with
one-stage and two-stage
cluster samples to estimate a population mean and a population proportion.

The Variability of the Estimate

The precision of a
sample design is directly related to the variability of the
estimate, which is measured by the
standard error. The tables below show how to compute
the standard error (SE),
when the sampling method is cluster sampling.

The first table shows how to compute the
standard error for a mean score, given one- or two-stage sampling.

The next table shows how to compute the standard error for a proportion. Like
the previous table, this table shows equations for one- and two-stage designs.
It also shows how the equations differ when the true population proportions are
known versus when they are estimated based on sample data.

Sample Problem

This section presents a sample problem that illustrates how to analyze survey
data when the sampling method is one-stage cluster sampling. (In a
subsequent lesson, we re-visit this problem and see how cluster
sampling compares to other sampling methods.)

Sample Planning Wizard

The analysis of data collected via cluster sampling can be complex and
time-consuming. Stat Trek's Sample Planning Wizard can help. The Wizard computes
survey precision, sample size requirements, costs, etc., as well as estimates
population parameters and tests hypotheses. It also creates a summary report
that lists key findings and documents analytical techniques. The Wizard
is free. You can find the Sample Planning Wizard in Stat Trek's
main menu under the Stat Tools tab. Or you can tap the button below.

At the end of every school year, the state administers a reading test to a
sample of third graders. The school system has 20,000 third graders, grouped in
1000 separate classes. Assume that each class has 20 students. This year, the
test was administered to each student in 36 randomly-sampled classes. Thus,
this is one-stage cluster sampling, with classes serving as clusters. The
average test score from each sampled cluster Xi
is shown below:

Find critical value. The critical value is a factor used to
compute the margin of error. Based on the
central limit theorem, we can assume that the
sampling distribution
of the mean is normally distributed. Therefore, we express the critical
value as a
z-score.
To find the critical value, we take these steps.